But an almighty hit by Muliaina on Bradley Davies knocked the ball out of the lock's grasp and set New Zealand away on another dangerous counter-attack.
The lesson of not giving this All Blacks side any loose ball to work with was hammered home once again in the 22nd minute, as they claimed a second try.
Carter fielded an aimless clearance and popped a simple ball to Muliaina on the scissors, allowing the full-back to race through almost untouched for an unconverted try.
By his high standards Carter was having an average day and minutes later the fly-half dallied a moment too long to clear, allowing Mike Phillips to block the kick and win a scrum in the Kiwi 22.
Wales probed left and then switched direction to create an overlap, only for James Hook to take on the chance himself and lose control as he slid for the line under Jimmy Cowan's tackle.
Their efforts deserved some reward, not least a powerful scrummaging performance that had the New Zealand front-row creaking.
While the clean break would not come, New Zealand were forced to ship penalties and Stephen Jones succeeded with two of the three chances presented to him - the one failure coming back off a post.
Coach Graham Henry hails 'composed' Kiwis
Carter, in contrast, was struggling with his kicking and another miss - his fourth failure of five - in first-half injury time left the All Blacks with just a 13-9 lead at the break.
Wales had the first sniff of the second period as they created space on the right for Tom Shanklin to race up field, but the move eventually broke down when wing Tom James was unable to hang onto a pass fractionally behind him.
New Zealand were rocking again when Rees emerged from a maul with the ball and kicked into space in the corner.
McCaw was on hand to gather, but was engulfed in chasers and Wales again ripped back possession to send a series of thrusts into the 22.
The All Blacks showed their class under pressure, however, and waited their moment to tear into a ruck in numbers to reclaim the ball.
Back came Wales and New Zealand finally lost their cool under pressure, replacement back-row Daniel Braid, on for the injured Kieran Read, earning a yellow card and presenting Jones with a regulation penalty to bring Wales to within a point.
Coach Warren Gatland 'proud' of Wales effort
Yet despite being a man down New Zealand replied with a classy try as Conrad Smith combined with Ma'a Nonu - on for Williams at centre - to bust the Welsh defence.
McCaw was in support to ensure quick ball and the All Blacks swept left for Gear to squeeze in at the flag despite George North's tackle.
Carter slotted the difficult conversion - radar ominously fixed - and New Zealand led 20-12 with 26 minutes remaining.
The fly-half added another penalty as New Zealand went back to basics by driving forward through the pack, but the Wales forwards were also hugely impressive at that type of game.
Jones was able to reply almost immediately thanks to his pack's efforts and then pulled Wales to within a converted try with a booming 47-metre kick.
But with a sniff of victory in their nostrils Wales were knocked back when the excellent Jerome Kaino caught Hook as he tried to kick, leading to a try for wing Toeava.
A fifth New Zealand try followed when Cowan's pass behind the back wrong-footed the Welsh defence for replacement prop Afoa to go under the posts.
Wales had the last say as Byrne finally crossed the Kiwi line after Shanklin had again shown up well, but Wales' long wait for a win over the All Blacks continues.
Wales: Byrne, North, Shanklin, Hook, Tom James, Stephen Jones, Phillips, Jenkins, Matthew Rees, Adam Jones, Davies, Alun Wyn Jones, Lydiate (Powell 48), Warburton (Martyn Williams 75), Ryan Jones (Thomas 48). Subs Not Used: Bennett, Paul James, Richie Rees, Bishop.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.